It is generally known that, in a hydraulic system with a main fluid-flow passage for feeding oil (lubricating oil) to moving parts of an internal combustion engine requiring lubrication and a branch passage branched from the main flow passage for feeding oil (working oil) to a hydraulic actuator, a control valve apparatus is disposed in the main flow passage downstream of a branched point of the branch passage from the main flow passage, for controlling a flow rate of oil flowing through the main flow passage downstream of the branched point. One such control valve apparatus has been disclosed in Japanese Patent Provisional Publication No. 57-173513 (hereinafter is referred to as “JP57-173513”), corresponding to U.S. Pat. No. 4,452,188, issued on Jun. 5, 1984. In the control valve apparatus (the oil-feed control apparatus) disclosed in JP57-173513, when the flow rate of oil fed into the main flow passage is limited during operation of an internal combustion engine at low speeds, the flow rate of oil flowing through the main flow passage downstream of the branched point is controlled to a small amount by means of the control valve apparatus. As a result of this, oil can be preferentially fed into the branch passage, thereby enhancing the responsiveness of a hydraulic actuator, to which oil (working oil) is delivered by way of the branch passage.